In this section we state the main results. Results pertaining to persistence of normally hyperbolic critical manifolds as locally invariant slow manifolds, as well as persistence of the stable/unstable manifolds (REF ) and their locally invariant foliations are given. For expository reasons we have decided to state the results via a series of independent statements, similarly to the presentation of Fenichel's theorems for fast-slow ODEs in [1]}. We shall also – again similarly again to [1]} – present two versions of the result describing perturbations of the critical manifold, the latter being a specialisation to the case in which the obtained slow manifold/foliation has a graph representation. Though the latter (graph) formulation is less general and easily derived from the former (manifold) formulation, it is worthwhile to present both here for two reasons. Firstly, the critical manifold \(S\)  is frequently a graph in applications. Secondly, a graph formulation is always achievable locally. This fact is also leveraged in the proofs in Section , where we shall typically restrict to the analysis of local graph formulations since the more global statements for compact manifolds are obtained via standard arguments based on a partition of unity.
